A195284 -id:A195284 - cp's OEIS frontend

A195381 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(2,sqrt(5),3).

1, 3, 2, 3, 1, 6, 9, 0, 7, 6, 4, 9, 9, 2, 1, 4, 9, 9, 5, 4, 0, 3, 0, 7, 3, 6, 2, 4, 7, 3, 5, 2, 1, 7, 4, 8, 9, 9, 9, 5, 4, 9, 4, 0, 5, 6, 1, 3, 9, 5, 5, 1, 0, 5, 7, 5, 7, 9, 8, 4, 7, 1, 7, 2, 2, 4, 2, 3, 1, 5, 9, 5, 8, 7, 8, 9, 4, 2, 1, 0, 7, 7, 7, 2, 4, 1, 5, 1, 1, 8, 3, 4, 1, 3, 0, 7, 2, 2, 0, 9

Offset: 1

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			(A)=1.32316907649921499540307362473521748999...

Cf. A195284, A195383, A195384, A195385.

Sqrt(12) / ((1 + Sqrt(5)) / 2)^2; // Vincenzo Librandi, Nov 15 2018

a = 2; b = Sqrt[5]; c = 3; f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%]   (* (A) A195381 *)
N[x2, 100]
RealDigits[%]   (* (B) A195383 *)
N[x3, 100]
RealDigits[%]   (* (C) A195384 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]   (* Philo(ABC,I) A195385 *)
RealDigits[Sqrt[12] / ((1 + Sqrt[5]) / 2)^2, 10, 100] (* Vincenzo Librandi, Nov 15 2018 *)

Equals sqrt(12)/phi^2, where phi = A001622. - Jon Maiga, Nov 14 2018

A195383 Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(2,sqrt(5),3).

Original entry on oeis.org

1, 3, 5, 4, 0, 4, 4, 6, 2, 7, 7, 7, 2, 8, 4, 5, 8, 7, 1, 2, 8, 3, 3, 4, 4, 5, 0, 9, 1, 0, 4, 2, 8, 7, 1, 2, 4, 0, 6, 0, 4, 5, 8, 0, 9, 0, 6, 6, 0, 7, 0, 3, 6, 1, 9, 9, 7, 8, 9, 0, 3, 6, 6, 7, 7, 8, 5, 9, 7, 3, 8, 2, 3, 2, 1, 1, 8, 6, 9, 5, 5, 8, 9, 3, 8, 1, 4, 2, 5, 6, 0, 7, 7, 6, 8, 9, 8, 9, 8, 3

Offset: 1

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			(B)=1.354044627772845871283344509104287124060458090...

Cf. A195284.

a = 2; b = Sqrt[5]; c = 3; f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%]   (* (A) A195381 *)
N[x2, 100]
RealDigits[%]   (* (B) A195383 *)
N[x3, 100]
RealDigits[%]   (* (C) A195384 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]   (* Philo(ABC,I) A195385 *)

A195384 Decimal expansion of shortest length, (C), of segment from side CA through incenter to side CB in right triangle ABC with sidelengths (a,b,c)=(2,sqrt(5),3).

Original entry on oeis.org

1, 7, 4, 8, 0, 6, 4, 0, 9, 7, 7, 9, 5, 2, 8, 4, 2, 8, 3, 1, 9, 7, 2, 0, 4, 8, 2, 0, 2, 2, 3, 0, 2, 0, 4, 5, 5, 1, 4, 9, 8, 8, 3, 2, 6, 3, 9, 4, 8, 2, 6, 8, 7, 5, 3, 6, 8, 0, 8, 2, 5, 1, 1, 4, 8, 0, 1, 8, 6, 1, 9, 6, 0, 1, 7, 7, 1, 3, 1, 1, 8, 2, 4, 9, 3, 8, 6, 0, 5, 7, 4, 0, 5, 1, 6, 5, 8, 7, 2, 2

Offset: 1

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			(C)=1.74806409779528428319720482022302045514988...

Cf. A195284.

a = 2; b = Sqrt[5]; c = 3; f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%]   (* (A) A195381 *)
N[x2, 100]
RealDigits[%]   (* (B) A195383 *)
N[x3, 100]
RealDigits[%]   (* (C) A195384 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]   (* Philo(ABC,I) A195385 *)

A195385 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 2,sqrt(5),3 right triangle ABC.

Original entry on oeis.org

6, 1, 1, 5, 5, 8, 3, 5, 1, 2, 7, 9, 6, 6, 4, 1, 3, 0, 6, 5, 6, 7, 7, 9, 3, 2, 2, 4, 2, 8, 8, 4, 4, 3, 2, 7, 0, 7, 7, 7, 0, 6, 7, 2, 5, 0, 0, 1, 8, 5, 3, 0, 9, 3, 1, 9, 2, 6, 0, 2, 3, 8, 0, 2, 9, 1, 7, 4, 6, 7, 0, 8, 6, 0, 9, 1, 9, 8, 1, 4, 4, 8, 1, 2, 6, 1, 1, 2, 9, 5, 1, 3, 1, 2, 6, 9, 9, 1, 5, 3, 7, 8, 6, 5, 4

Offset: 0

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			Philo(ABC,I)=0.6115583512796641306567793224288443270777...

Cf. A195284.

a = 2; b = Sqrt[5]; c = 3; f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%]   (* (A) A195381 *)
N[x2, 100]
RealDigits[%]   (* (B) A195383 *)
N[x3, 100]
RealDigits[%]   (* (C) A195384 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]   (* Philo(ABC,I) A195385 *)

a(99) corrected by Georg Fischer, Jul 18 2021

A195386 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(sqrt(2),sqrt(5),sqrt(7)).

Original entry on oeis.org

1, 0, 4, 5, 8, 3, 1, 3, 7, 9, 9, 7, 9, 9, 5, 5, 8, 7, 4, 9, 4, 8, 7, 2, 0, 5, 7, 5, 7, 0, 3, 4, 1, 1, 6, 8, 1, 4, 2, 4, 8, 5, 2, 0, 4, 7, 4, 4, 8, 0, 2, 4, 4, 0, 9, 4, 4, 5, 3, 8, 9, 4, 5, 8, 9, 0, 4, 0, 7, 2, 1, 2, 7, 2, 0, 5, 8, 6, 7, 2, 9, 0, 3, 5, 6, 3, 1, 8, 0, 3, 1, 7, 9, 4, 4, 5, 7, 4, 1, 1

Offset: 1

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			(A)=1.0458313799799558749487205757034116814248520474480...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A195284, A195387, A195388, A195389.

a = Sqrt[2]; b = Sqrt[5]; c = Sqrt[7];
f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%]   (* (A) A195386 *)
N[x2, 100]
RealDigits[%]   (* (A) A195387 *)
N[x3, 100]
RealDigits[%]   (* (A) A195388 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]   (* Philo(ABC,I) A195389 *)

A195387 Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(sqrt(2),sqrt(5),sqrt(7)).

Original entry on oeis.org

1, 1, 4, 6, 8, 0, 9, 7, 5, 9, 1, 5, 8, 1, 9, 1, 6, 3, 0, 9, 5, 3, 7, 7, 6, 0, 0, 6, 5, 1, 9, 6, 8, 1, 6, 0, 7, 5, 5, 6, 7, 6, 8, 2, 9, 7, 3, 5, 9, 7, 5, 1, 3, 7, 2, 7, 2, 9, 8, 2, 4, 8, 5, 3, 3, 1, 7, 8, 9, 4, 6, 4, 4, 3, 9, 9, 1, 8, 6, 0, 9, 3, 6, 7, 6, 2, 0, 5, 1, 5, 2, 1, 5, 4, 4, 9, 5, 5, 0, 7

Offset: 1

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			(B)=1.1468097591581916309537760065196816075567682...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A195284, A195386, A195388, A195389.

a = Sqrt[2]; b = Sqrt[5]; c = Sqrt[7];
f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%]   (* (A) A195386 *)
N[x2, 100]
RealDigits[%]   (* (A) A195387 *)
N[x3, 100]
RealDigits[%]   (* (A) A195388 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]   (* Philo(ABC,I) A195389 *)

A195388 Decimal expansion of shortest length, (C), of segment from side CA through incenter to side CB in right triangle ABC with sidelengths (a,b,c)=(sqrt(2),sqrt(5),sqrt(7)).

Original entry on oeis.org

1, 4, 2, 0, 6, 2, 0, 2, 7, 3, 3, 9, 4, 4, 3, 7, 9, 4, 6, 4, 1, 5, 1, 4, 4, 8, 1, 2, 1, 1, 6, 1, 6, 9, 2, 3, 1, 9, 6, 3, 5, 3, 5, 3, 3, 1, 5, 4, 6, 4, 8, 9, 8, 8, 0, 5, 5, 3, 7, 5, 9, 3, 8, 5, 4, 7, 2, 5, 5, 9, 2, 8, 2, 3, 3, 2, 2, 9, 9, 1, 9, 3, 3, 6, 7, 4, 3, 8, 2, 1, 3, 1, 8, 4, 9, 2, 0, 7, 2, 3

Offset: 1

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			(C)=1.420620273394437946415144812116169231963535...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A195284, A195386, A195387, A195389.

a = Sqrt[2]; b = Sqrt[5]; c = Sqrt[7];
f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%]   (* (A) A195386 *)
N[x2, 100]
RealDigits[%]   (* (A) A195387 *)
N[x3, 100]
RealDigits[%]   (* (A) A195388 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]   (* Philo(ABC,I) A195389 *)

A195389 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a sqrt(2),sqrt(5),sqrt(7) right triangle ABC.

Original entry on oeis.org

5, 7, 3, 8, 9, 4, 9, 4, 2, 7, 4, 8, 6, 8, 2, 3, 0, 6, 8, 5, 9, 4, 1, 0, 2, 1, 1, 4, 2, 6, 4, 4, 0, 2, 2, 8, 6, 9, 3, 9, 8, 0, 8, 1, 9, 5, 3, 5, 4, 9, 9, 1, 1, 5, 0, 5, 7, 5, 2, 0, 9, 5, 2, 0, 9, 2, 4, 5, 4, 7, 0, 8, 0, 9, 5, 1, 8, 9, 1, 7, 5, 0, 5, 0, 8, 1, 2, 7, 6, 3, 1, 2, 8, 9, 1, 0, 5, 0, 7, 7

Offset: 0

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			Philo(ABC,I)=0.5738949427486823068594102114264402286939808...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A195284, A195386, A195387, A195388.

a = Sqrt[2]; b = Sqrt[5]; c = Sqrt[7];
f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%]   (* (A) A195386 *)
N[x2, 100]
RealDigits[%]   (* (A) A195387 *)
N[x3, 100]
RealDigits[%]   (* (A) A195388 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]   (* Philo(ABC,I) A195389 *)

A195395 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(sqrt(3),sqrt(5),sqrt(8)).

Original entry on oeis.org

1, 2, 0, 4, 4, 9, 9, 9, 5, 2, 4, 3, 8, 3, 0, 0, 4, 2, 2, 9, 6, 2, 6, 7, 7, 2, 0, 4, 9, 5, 5, 8, 8, 0, 4, 2, 5, 3, 7, 2, 4, 9, 9, 8, 3, 8, 1, 4, 3, 2, 7, 9, 8, 3, 2, 8, 9, 2, 3, 7, 3, 3, 6, 2, 4, 6, 2, 0, 5, 8, 0, 7, 9, 0, 1, 7, 0, 6, 1, 9, 5, 8, 9, 3, 3, 1, 3, 9, 8, 9, 3, 0, 0, 9, 4, 1, 9, 1, 5, 1

Offset: 1

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			(A)=1.204499952438300422962677204955880425372499...

Cf. A195284, A195896, A195897, A195398.

a = Sqrt[3]; b = Sqrt[5]; c = Sqrt[8];
f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%]    (* (A) A195395 *)
N[x2, 100]
RealDigits[%]    (* (B) A195396 *)
N[x3, 100]
RealDigits[%]    (* (C) A195397 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]    (*  Philo(ABC,I) A195398 *)

A195396 Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(sqrt(3),sqrt(5),sqrt(8)).

Original entry on oeis.org

1, 2, 6, 9, 3, 1, 5, 7, 9, 8, 8, 6, 2, 5, 6, 0, 6, 6, 9, 2, 8, 7, 2, 7, 6, 7, 3, 2, 7, 3, 8, 9, 4, 5, 3, 9, 8, 4, 5, 1, 4, 1, 2, 8, 2, 1, 3, 5, 8, 1, 0, 2, 7, 4, 6, 3, 2, 9, 7, 6, 8, 8, 0, 1, 3, 5, 3, 3, 3, 4, 3, 2, 3, 8, 8, 1, 6, 1, 5, 3, 8, 4, 7, 1, 0, 3, 8, 3, 9, 2, 5, 9, 5, 2, 6, 3, 5, 2, 0, 7

Offset: 1

Clark Kimberling, Sep 17 2011

See A195284 for definitions and a general discussion.

			(B)=1.2693157988625606692872767327389453984514...

Cf. A195284.

a = Sqrt[3]; b = Sqrt[5]; c = Sqrt[8];
f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%]    (* (A) A195395 *)
N[x2, 100]
RealDigits[%]    (* (B) A195396 *)
N[x3, 100]
RealDigits[%]    (* (C) A195397 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]    (*  Philo(ABC,I) A195398 *)

cp's OEIS Frontend

A195381 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(2,sqrt(5),3).

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A195383 Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(2,sqrt(5),3).

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A195384 Decimal expansion of shortest length, (C), of segment from side CA through incenter to side CB in right triangle ABC with sidelengths (a,b,c)=(2,sqrt(5),3).

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A195385 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 2,sqrt(5),3 right triangle ABC.

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A195386 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(sqrt(2),sqrt(5),sqrt(7)).

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Mathematica

A195387 Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(sqrt(2),sqrt(5),sqrt(7)).

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Comments

Examples

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Crossrefs

Programs

Mathematica

A195388 Decimal expansion of shortest length, (C), of segment from side CA through incenter to side CB in right triangle ABC with sidelengths (a,b,c)=(sqrt(2),sqrt(5),sqrt(7)).

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

A195389 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a sqrt(2),sqrt(5),sqrt(7) right triangle ABC.

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Author

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Examples

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Programs

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